IJPAM: Volume 53, No. 2 (2009)

FORMULAE RELATIONSHIP TO THE DISTRIBUTIONAL
PRODUCT OF $\delta ^{(k-1)}(x).\delta ^{(l-1)}(x)$ AND $\delta ^{(k-1)}(x-a).\delta ^{(l-1)}(x-b)$

Manuel A. Aguirre
Núcleo Consolidado Matemática Pura y Aplicada
Facultad de Ciencias Exactas
Universidad Nacional del Centro
Tandil, Provincia de Buenos Aires, ARGENTINA
e-mail: maguirre@exa.unicen.edu.ar

Abstract.One of the problem in distribution theory is the lack of definitions for products and power of distributions in general. In physics (c.f. [#!Ga!#], p. 141), oneself finds the need to evaluate $\delta ^{2}$ when calculating the transition rates of certain particle interactions. Chenkuan Li (see [#!L!#]) derives that $\delta ^{2}(x)=0$ on even-dimension space by applying the Laurent expansion of . Koh and Li in [#!K!#] give a sense to distribution $\delta ^{k}$ and $(\delta ^{\lq })^{k}$ for some , using the concept of neutrix limit. Aguirre in [#!A!#], gives a sense to distributional product of $\delta ^{(m)}(x).\delta ^{(l)}(x)$ , using the Hankel transform of generalized function of $\delta ^{(m)}(x)$ . In this paper using the Fourier transform of $\delta ^{(k-1)}(x)$ we obtain formulae for the distributional product of $\delta ^{(k-1)}(x).\delta ^{(l-1)}(x)$ and $% \delta ^{(k-1)}(x-a). \delta ^{(l-1)}(x-b),$ where and . As consequence we give a sense at the following product: $\delta ^{2}(x),(\delta (x))^{k},(\delta ^{(k-1)}(x))^{m},(\delta (x-a))^{t}$ and $% (\delta ^{k}(x-a))^{m}$ . Finally, we write formulae relations with distributional products of $(\delta ^{k}(P(x_{1},. . . x_{n}))^{2}$ and $% (\delta ^{k}(m^{2}+P(x_{1},. . . x_{n}))^{2}$ where $P(x_{1},. . . x_{n})$ is defined by ().